The formula is important in the trace theory for Sobolev spaces, since the left-hand side of (REF ) is the classical Dirichlet integral and the right-hand side is equivalent to the Gagliardo form in \(H^{1/2}(\partial B(0,1))\) , the trace space for \(W^{1,2}(B(0,1))\) . The identity inspired important developments in the theory of Dirichlet forms; see [1]}, [2]}, [3]}.
Doob [4]} generalized (REF )
to arbitrary Greenian open sets \(D\subseteq \mathbb {R}^d\)  with \(d\ge 2\) .
In this paper, we propose another extension of (REF ):
\(\int \limits _{B(0,1)} |\nabla u(x)|^2 |u(x)|^{p-2}\,dx = \frac{1}{2 (p-1)} \int _0^{2\pi } \int _0^{2\pi } \frac{(g(\eta )^{\langle p-1 \rangle } - g(\xi )^{\langle p-1 \rangle })(g(\eta ) - g(\xi ))}{4\pi \sin ^2((\xi -\eta )/2)}\,d\eta \, d\xi ,\) 
